1,2

a(n) occurs in the process of normal ordering of the n-th power of a product of the cubes of the boson creation and boson annihilation operators.

a(11)=110264570238241604072673394 =~ 10^26.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.

M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem.

a(n)= sum((((k+3)!)^n)/((k+3)!*(k!)^n), k=0..infinity)/exp(1), n>=1. This is a Dobinski-type summation formula.

a(n)= (sum(((k*(k-1)*(k-2))^n)/k!, k=3..infinity)/exp(1), n>=1. Usually a(0) := 1. (From eq.(26) with r=3 of the Schork reference; rewritten original eq.(25) with r=3 of the Blasiak et al. reference.)

E.g.f. with a(0) := 1: (sum((exp(k*(k-1)*(k-2)*x))/k!, k=3..infinity)+5/2)/exp(1). From top of p. 4656 with r=3 of the Schork reference.

f[n_] := f[n] = Sum[(k + 3)!^n/((k + 3)!*(k!^n)*E), {k, 0, Infinity}]; Table[ f[n], {n, 1, 9}]

Cf. A000110 and A020556, if k+3 is replaced by k+1 or k+2, respectively.

Sequence in context: A086881 A056566 A138590 this_sequence A129056 A005334 A033511

Adjacent sequences: A069220 A069221 A069222 this_sequence A069224 A069225 A069226

nonn,easy

Karol A. Penson (penson(AT)lptl.jussieu.fr), Apr 12 2002

Edited by Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 30 2002